Finite-density massless two-color QCD at the isospin Roberge-Weiss point and the 't Hooft anomaly

We study the phase diagram of two-flavor massless two-color QCD (${\mathrm{QC}}_2\mathrm{D}$) under the presence of quark chemical potentials and imaginary isospin chemical potentials. At the special point of the imaginary isospin chemical potential, called the isospin Roberge-Weiss (RW) point, two-flavor ${\mathrm{QC}}_2\mathrm{D}$ enjoys the ${\mathbb{Z}}_2$ center symmetry that acts on both quark flavors and the Polyakov loop. We find a ${\mathbb{Z}}_2$ 't Hooft anomaly of this system, which involves the ${\mathbb{Z}}_2$ center symmetry, the baryon-number symmetry, and the isospin chiral symmetry. Anomaly matching, therefore, constrains the possible phase diagram at any temperatures and quark chemical potentials at the isospin RW point, and we compare it with previous results obtained by chiral effective field theory and lattice simulations. We also point out an interesting similarity of two-flavor massless ${\mathrm{QC}}_2\mathrm{D}$ with ($2+1$)-dimensional quantum antiferromagnetic systems.

Figure 1

(a) A schematic phase diagram of two-flavor massive ${\mathrm{QC}}_2\mathrm{D}$ with the thermal boundary condition, predicted by theoretical and numerical works. Since this theory has neither the center nor chiral symmetry, the boundaries of phases expressed by dashed curves will be a crossover. (b) A possible phase diagram for two-flavor massless ${\mathrm{QC}}_2\mathrm{D}$ at the isospin RW point, which we give in this work. Here we discuss the spontaneous symmetry breaking of the center, chiral, and baryon-number symmetries and classify the phases. Each boundary of phases is expected to be a critical line associated with a second-order phase transition.

Figure 2

Illustrations of the phases of the easy-plane $\mathbb{C}{P}^1$ model. (a) In the easy-axis Néel phase, the microscopic spins are ordered in a staggered manner and point in the $\pm z$ directions. (b) In the easy-plane Néel phase, the spins form an antiferromagnetic order, preferring the $xy$ plane. (c) In the valence bond solid phase, the spins are paired with their nearest neighbors to be spin-singlet. Green ellipses indicate the spin-singlet pairs.